Brane Tilings and Exceptional Collections

TL;DR

The paper builds a dictionary between brane tilings and exceptional collections for toric Calabi–Yau threefolds partially resolvable by a surface, enabling translation between quiver/gauge data and geometric data. It develops two complementary constructions: (i) deriving a periodic Beilinson quiver from an exceptional collection using Wilson lines on a torus, and (ii) extracting an exceptional collection from a brane tiling via internal perfect matchings and the $\Psi$-map. The work demonstrates that Beilinson quivers and tiling duals share the same connectivity and that Euler characteristics vanish consistently in this framework, providing a topological bridge between the two viewpoints. While not a full general proof of equivalence, the authors present concrete procedures, case studies (e.g., ${\mathbb P}^2$, ${\bf dP}_1$, and $Y^{3,2}$), and a clear roadmap for extending the correspondence to broader classes of toric CY singularities.

Abstract

Both brane tilings and exceptional collections are useful tools for describing the low energy gauge theory on a stack of D3-branes probing a Calabi-Yau singularity. We provide a dictionary that translates between these two heretofore unconnected languages. Given a brane tiling, we compute an exceptional collection of line bundles associated to the base of the non-compact Calabi-Yau threefold. Given an exceptional collection, we derive the periodic quiver of the gauge theory which is the graph theoretic dual of the brane tiling. Our results give new insight to the construction of quiver theories and their relation to geometry.

Figures (37)

• Figure 1: Quiver of ${\bf dP}_1$. The theory contains four $U(N)$ gauge groups labeled by the nodes of the quiver. The arrows label bifundamental fields transforming in the (anti--)fundamental representation of the groups at the endpoints.
• Figure 2: ${\bf dP}_1$ Beilinson quiver.
• Figure 3: The cone for the variety. The coordinates of the spanning vectors are integers. The endpoints are coplanar following from the Calabi--Yau condition.
• Figure 4: The toric diagram for $L^{1,7,3}$ which is part of the recently discovered series of $L^{abc}$ metrics (Cvetic:2005ftCvetic:2005vk). The dual quiver theories have been constructed in Franco:2005smBenvenuti:2005jaButti:2005sw.
• Figure 5: The ${\mathbb P}^2$ periodic quiver. The nodes denote $U(N)$ gauge groups; the directed edges between them are bifundamental fields. The plaquettes of the quiver graph are terms in the superpotential. This example has three gauge groups, labeled by numbers. Identifying nodes with the same labels (i.e. "compactifying" the periodic quiver) yields the usual quiver diagram.

Theorems & Definitions (9)

• Definition 5.1.1
• Definition 5.1.2
• Definition 5.1.3
• Lemma 5.1.4
• proof
• Lemma 5.3.1
• proof
• Corollary 5.3.2
• Corollary 5.3.3